Measurement sequence determination for quantum computing device

ABSTRACT

A computing system is provided, including a processor configured to identify a plurality of measurement sequences that implement a logic gate. Each measurement sequence may include a plurality of measurements of a quantum state of a topological quantum computing device. The processor may be further configured to determine a respective estimated total resource cost of each measurement sequence of the plurality of measurement sequences. The processor may be further configured to determine a first measurement sequence that has a lowest estimated total resource cost of the plurality of measurement sequences. The topological quantum computing device may be configured to implement the logic gate by applying the first measurement sequence to the quantum state.

BACKGROUND

Recent experiments have established the existence of Majorana zero modes(MZMs) in hybrid semiconductor-superconductor heterostructures. MZMshave been investigated as a platform for realizing topological quantumcomputation. A current line of research into topological quantumcomputing using MZMs aims to assemble a network of topologicalsuperconductors in a way that allows practical quantum informationprocessing on many qubits.

SUMMARY

According to one aspect of the present disclosure, a computing system isprovided, including a processor configured to identify a plurality ofmeasurement sequences that implement a logic gate. Each measurementsequence may include a plurality of measurements of a quantum state of atopological quantum computing device. The processor may be furtherconfigured to determine a respective estimated total resource cost ofeach measurement sequence of the plurality of measurement sequences. Theprocessor may be further configured to determine a first measurementsequence that has a lowest estimated total resource cost of theplurality of measurement sequences. The topological quantum computingdevice may be configured to implement the logic gate by applying thefirst measurement sequence to the quantum state.

This Summary is provided to introduce a selection of concepts in asimplified form that are further described below in the DetailedDescription. This Summary is not intended to identify key features oressential features of the claimed subject matter, nor is it intended tobe used to limit the scope of the claimed subject matter. Furthermore,the claimed subject matter is not limited to implementations that solveany or all disadvantages noted in any part of this disclosure.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows an example quantum computing system including a processorand a topological quantum computing device, the topological quantumcomputing device including two Majorana hexons, according to oneembodiment of the present disclosure.

FIG. 2 shows an example two-sided Majorana hexon architecture, accordingto the embodiment of FIG. 1.

FIG. 3 shows an example one-sided Majorana hexon architecture, accordingto the embodiment of FIG. 1.

FIG. 4 shows an example quantum computing system including a processorand a topological quantum computing device, the topological quantumcomputing device including two Majorana tetrons, according to anotherembodiment of the present disclosure.

FIG. 5 shows projection operators for a Majorana hexon, according to theembodiment of FIG. 1.

FIG. 6 shows joint fermionic parity operators expressed in terms ofPauli matrices, according to the embodiment of FIG. 1.

FIG. 7 shows diagrammatical representations of Majorana hexons includingcomputational qubits and ancillary qubits, according to the embodimentof FIG. 1.

FIG. 8 shows a braiding transformation and example measurement sequencesfor a Majorana hexon, according to the embodiment of FIG. 1.

FIG. 9 shows diagrammatical representations of example measurementsequences of Majorana hexons, according to the embodiment of FIG. 1.

FIG. 10 shows ancillary projection operators for a Majorana hexon,according to the embodiment of FIG. 1.

FIG. 11 shows a projector sequence including ancillary projectionoperators, according to the embodiment of FIG. 1.

FIG. 12 shows a two-qubit projection operator for four Majorana zeromodes of a Majorana hexon, according to the embodiment of FIG. 1.

FIG. 13 shows a measurement sequence that generates a two-qubit logicgate, according to the embodiment of FIG. 1.

FIGS. 14A-14C show example fermionic parity measurement configurationsin a two-sided Majorana hexon architecture, according to the embodimentof FIG. 2.

FIG. 14D shows an estimated weighted measurement cost for a fermionicparity measurement, according to the embodiment of FIG. 2.

FIGS. 15A-15E show example fermionic parity measurement configurationsin a one-sided Majorana hexon architecture, according to the embodimentof FIG. 3.

FIG. 16A shows example post-measurement states of a Majorana hexon,according to the embodiment of FIG. 1.

FIG. 16B shows a method for performing a forced measurement of aprojection operator, according to the embodiment of FIG. 1.

FIG. 17A shows a diagram of a projection operator sequence, according tothe embodiment of FIG. 1.

FIG. 17B shows a diagram of a projection operator sequence after ameasurement of an undesired parity of a projection operator has beenperformed, according to the embodiment of FIG. 17A.

FIG. 17C shows a diagram of a forced measurement applied to theprojection operator sequence of FIG. 17B.

FIG. 18 shows an estimated total resource cost for a measurementsequence in which a forced measurement occurs, according to theembodiment of FIGS. 16A-B.

FIG. 19A-19B show diagrams of example measurement sequences in which analternative forced measurement protocol is applied to the quantum state,according to the embodiment of FIG. 1.

FIG. 20 shows an estimated total resource cost for a measurementsequence in which the alternative forced measurement protocol isapplied, according to the embodiment of FIGS. 19A-B.

FIG. 21 shows an example measurement sequence that occurs when forcedmeasurement is used on four Majorana zero modes of a Majorana hexon,according to the embodiment of FIG. 1.

FIG. 22 shows the estimated total resource costs of measurementsequences including forced measurements of four Majorana zero modes of aMajorana hexon, according to the embodiment of FIG. 21.

FIG. 23 shows joint fermionic parity operators expressed in terms ofPauli operators, according to the embodiment of FIG. 1.

FIGS. 24A-24B show diagrams of measurement sequences that differ only byoverall Pauli operators, according to the embodiment of FIG. 23.

FIG. 25 shows the construction of a measurement sequence that compilesto a logic gate for a plurality of Majorana hexons, according to theembodiment of FIG. 1.

FIG. 26 shows the estimated total resource cost of the measurementsequence of FIG. 25 when a Majorana-Pauli tracking protocol is used.

FIG. 27 shows the estimated total resource cost of the measurementsequence of FIG. 25 when a forced measurement protocol is used.

FIG. 28 shows a flowchart of an example method that may be used forperforming a quantum computation by implementing a logic gate, accordingto the embodiment of FIG. 1.

FIG. 29 shows a schematic view of an example computing environment inwhich the quantum computing system of FIG. 1 may be enacted.

DETAILED DESCRIPTION

Measurement-only topological quantum computation is one approach totopological quantum computation that is well-suited to implementationusing Majorana zero modes (MZMs). Measurement-only topological quantumcomputation allows computations to be performed without physicallymoving the MZMs, which are typically bound to macroscopic defects (suchas the ends of wires, as discussed in further detail below) and may bedifficult to move. Instead, braiding transformations may be performedthrough a series of (potentially non-local) measurements on sets of MZMsinvolving the MZMs that encode the computational state that is to bemanipulated and another set of MZMs that serve as ancillary degrees offreedom. The MZMs may, in some architectures, be coupled to quantumdots, thus allowing the states of the MZMs to be measured by measuringthe effects of the MZMs on the energy spectra of the quantum dots.

Using MZMs, a Clifford gate (for example, a Hadamard gate, a π/4 phasegate, or a controlled not gate) may be constructed in a quantumcomputing device. In order to perform universal quantum computation, anadditional gate such as a T-gate (a π/8 phase gate) may additionally beimplemented. The Clifford gate may be topologically protected such thatperturbations to the quantum state are suppressed when the Clifford gateis implemented. The T-gate may not be topologically protected in suchconfigurations.

Each Clifford gate implemented at the quantum computing device may becompiled from a sequence of measurements of the quantum state. Accordingto previous approaches for constructing a Clifford gate from a sequenceof measurements, a measurement sequence with minimal length is generatedfor each of the basic braiding transformations for each qubit. In suchapproaches, a minimal-length measurement sequence for a two-qubitentangling gate is then generated for each pair of qubits, and theresulting gate set is used as a generating gate set to synthesize anyother Clifford gates. However, this approach may be inefficient, sincethere may exist shorter sequences of measurements that compile to thesame gate.

In order to address the above inefficiency of existing methods forcompiling Clifford gates in topological quantum computing devices, acomputing system 10 is provided, as shown in the example embodiment ofFIG. 1. The computing system 10 may include a processor 12 and memory14, which may be operatively coupled. The processor 12 and memory 14 maybe included in a classical computing device. The computing system 10 mayfurther include other hardware components such as one or more inputdevices, one or more output devices, and/or one or more communicationdevices. In some embodiments, the functions of the computing system 10may be distributed across a plurality of communicatively coupledcomputing devices.

As discussed in further detail below, the processor 12 may be configuredto identify a plurality of measurement sequences 50 that implement alogic gate 40. The logic gate 40 may be a one-qubit Clifford gate or amulti-qubit Clifford gate. In some embodiments, as shown in the exampleof FIG. 1, the computing system 10 may include a topological quantumcomputing device 20 having a quantum state 22. Each measurement sequence50 of the plurality of measurement sequences 50 may include a pluralityof measurements 52 of the quantum state 22 of the topological quantumcomputing device 20. In such embodiments, the processor 12 may beoperatively coupled to the topological quantum computing device 20 andmay be configured to transmit one or more measurement sequences 50 tothe topological quantum computing device 50 to implement the one or morelogic gates 40. In other embodiments, the processor 12 may be configuredto identify the plurality of measurement sequences 50 for later use witha topological quantum computing device 20 when the computing system 10does not include a topological quantum computing device 20.

The processor 12 may be further configured to determine a respectiveestimated total resource cost 56 of each measurement sequence 50 of theplurality of measurement sequences 50. The estimated total resource cost56 of a measurement sequence 50 may, for example, indicate an amount oftime or an amount of energy that implementing the logic gate 40 with themeasurement sequence 50 is estimated to consume. Additionally oralternatively, the estimated total resource cost 56 may indicate anerror rate of the measurement sequence 50.

In some embodiments, for each measurement sequence 50 of the pluralityof measurement sequences 50, the processor 12 may be configured todetermine the respective estimated total resource cost 56 at least inpart by determining an estimated weighted resource cost 54 of eachmeasurement 52 included in the measurement sequence 50. For example, theestimated weighted resource cost 54 of each measurement 52 may indicatean error rate of the measurement 52. The processor 12 may then befurther configured to determine the estimated total resource cost 56 ofthe measurement sequence 50 based on the plurality of estimated weightedresource costs 54 of the individual measurements 52. For example, theestimated total resource cost 56 may be the product of the estimatedweighted resource costs 54.

Once the respective estimated total resource costs 56 of the pluralityof measurement sequences 50 have been determined, the processor 12 maybe further configured to determine a first measurement sequence 60 thathas a lowest estimated total resource cost 66 of the plurality ofmeasurement sequences 50. The processor 12 may, in some embodiments, befurther configured to transmit the first measurement sequence 60 to thetopological quantum computing device 20 so that the topological quantumcomputing device 20 may implement the logic gate 40 by applying thefirst measurement sequence 60 to the quantum state 22.

The topological quantum computing device 20 may include one or moreMajorana hexons, as shown in FIG. 1. The example of FIG. 1 shows a firstMajorana hexon 30A. In some embodiments, the topological quantumcomputing device 20 may further include a second Majorana hexon 30B.Each Majorana hexon may include six Majorana zero modes (MZMs). Thefirst Majorana hexon 30A includes Majorana zero modes 32A, 32B, 32C,32D, 32E, and 32F. The second Majorana hexon 30B includes Majorana zeromodes 32G, 32H, 32I, 32J, 32K, and 32L. In some embodiments, thetopological quantum computing device 20 may include more than twoMajorana hexons.

Example structures that may be included in the topological quantumcomputing device 20 to implement one or more qubits are shown in FIGS. 2and 3. The topological quantum computing device 20 may instantiate theplurality of MZMs 32 in a one-sided architecture or a two-sidedarchitecture. FIG. 2 shows an example of a two-sided Majorana hexonarchitecture 80. In the embodiment of FIG. 2, the MZMs 32 included ineach Majorana hexon 30 are coupled by a topological superconductor 70and a superconductor 72. Within each Majorana hexon 30, thesuperconductor 72 is located between a first side 72A on which threeMZMs 32 (labeled γ₁, γ₂, and γ₃) are located and a second side 72B onwhich the other three MZMs 32 (labeled γ₄, γ₅, and γ₆) are located. Thetwo-sided Majorana hexon architecture 80 of FIG. 2 includes a pluralityof Majorana hexons 30, and further includes a semiconductor 74 that islocated between adjacent Majorana hexons 30 and is coupled to thetopological superconductor 70. Each MZM 32 may form at a junctionbetween the topological superconductor 70 and the semiconductor 74. Whenthe two-sided Majorana hexon architecture 80 of FIG. 2 is used, eachMajorana hexon 30 may have a square lattice connectivity in which eachMajorana hexon 30 may be entangled with its vertical and horizontalneighbors.

An example a one-sided Majorana hexon architecture 82 is shown in theexample of FIG. 3. In the one-sided Majorana hexon architecture 82, eachMZM 32 included in a Majorana hexon 30 is located on the same side ofthe superconductor 72. As in the example embodiment of FIG. 2, each MZM32 may be formed at a junction between a topological superconductor 70and a semiconductor 74. FIG. 3 further shows a close-up view of aportion of the semiconductor 74 in the one-sided Majorana hexonarchitecture 82. In the example of FIG. 3, a plurality of quantum dots78 are embedded within the semiconductor 74 proximate the MZMs 32. Inaddition, the close-up view of the portion of the semiconductor 74 showsa plurality of cutter gates 76 located between the quantum dots 78 andthe MZMs 32 and between pairs of quantum dots 78. The cutter gates 76and the quantum dots 78 are described in further detail below.

In the one-sided Majorana hexon architecture 82 of FIG. 3, the pluralityof Majorana hexons 30 may have an asymmetric square lattice connectivityin which the one-sided Majorana hexon architecture 82 does not havevertical reflection symmetry. Thus, two-qubit gates acting onhorizontally displaced qubits may have different leftward and rightwardoperations. A two-qubit gate included in the two-sided Majorana hexonarchitecture 80 of FIG. 2 may also have different leftward and rightwardoperations if the labeling of the MZMs 32 is asymmetric under left-rightreflection.

In the two-sided Majorana hexon architecture 80 and the one-sidedMajorana hexon architecture 82, each Majorana hexon 30 may begalvanically isolated from other Majorana hexons 30. Thus, each Majoranahexon 30 may have a charging energy E_(C) resulting from Coulombinteractions. The charging energy of each Majorana hexon 30 may reducethe probability of quasiparticle poisoning of that Majorana hexon 30,since the probability for an electron to tunnel onto or off of theisland is exponentially suppressed in the ratio of the charging energyE_(C) to temperature, exp(−E_(C)/k_(B)T), where T is the temperature ofthe Majorana hexon 30 and k_(B) is Boltzmann's constant.

Projective measurements of the joint fermionic parity of any two MZMs 32may be performed by enabling weak coherent single-electron tunnelingbetween the MZMs 32 included in the pair and the quantum dots 78adjacent to those MZMs 32. This coupling gives rise to a shift in theenergy spectrum and charge occupation of the dot, which may then bemeasured. This measurement may be topologically protected in the sensethat the operator that is being measured is known up to corrections thatare exponentially small in the separation distance, through thesuperconductor 72 and the topological superconductor 70, of MZMs 32 inthe pair. However, measurement fidelity may be limited by thesignal-to-noise ratio and decoherence of the qubit.

In each Majorana hexon 30, a qubit may be formed from four MZMs 32. Theremaining two MZMs 32 included in the Majorana hexon 30 may be used asancillary MZMs 32 when performing measurement-based topologicaloperations, as discussed in further detail below. In other embodiments,additionally or alternatively to a Majorana hexon 30, the topologicalquantum computing device may include two Majorana tetrons. FIG. 4 showsan example computing system 100 including a topological quantumcomputing device 120 with a quantum state 122. The topological quantumcomputing device 120 of FIG. 4 includes two Majorana tetrons. In theembodiment of FIG. 4, one Majorana tetron is a computational Majoranatetron 130A that encodes a computational qubit and the other Majoranatetron is an ancillary Majorana tetron 130B that encodes an ancillaryqubit. The computational Majorana tetron 130A includes four Majoranazero modes 132A, 132B, 132C, and 132D. The ancillary Majorana tetron130B includes Majorana zero modes 132E, 132F, 132G, and 132H. In otherembodiments, the topological quantum computing device 120 may includesome other number of Majorana tetrons.

In other embodiments not shown in the figures, the topological quantumcomputing device 20 may include one or more Majorana octons that eachinclude eight MZMs 30.

Some measurements 52 of the quantum state 22 may be more difficult toperform than other measurements 52. As discussed above, the difficultyof performing a measurement 52 may be indicated by an estimated weightedresource cost 54 of that measurement 52. In the two-sided Majorana hexonarchitecture 80 of FIG. 2, some measurements 52 may be performed betweenMZMs 32 on the same side (left or right) of the Majorana hexon 30. Othermeasurements may be performed between MZMs 32 on opposite sides of theMajorana hexon 30. When the MZMs 32 have a greater separation distance,enabling coherent single-electron tunneling between these MZMs 32 and acommon quantum dot 78 may be more difficult, since the distance betweenthe MZMs 32 may exceed the phase coherence length of the semiconductor74. Thus, the estimated weighted resource cost 54 of a measurement 52may increase with increasing distance between the MZMs 32 for which themeasurement 52 is performed. In some embodiments, the Majorana hexon 30may include at least one coherent superconducting link between a pair ofnon-adjacent MZMs 32 in order to facilitate coherent single-electrontunneling between the non-adjacent MZMs 32.

As shown in the examples of FIGS. 2 and 3, a plurality of Majoranahexons 30 may be included in the topological quantum computing device20. Multi-qubit operations may be performed by weakly coupling MZMs 32from different Majorana hexons 30 to common quantum dots 78. Since thecoupling is weak, the charging energy protection against quasi-particlepoisoning may remain effective during such operations. The chargingenergy E_(C) of each Majorana hexon 30 may be maintained duringmeasurements 52 by performing measurements 52 of operators that commutewith the charging energy E_(C). These measurements 52 may be those thatinclude even numbers of Majorana operators, as discussed in furtherdetail below.

Each of the six MZMs 32 included in a Majorana hexon 30 may be labeledwith subscripts 1 through 6 and may each be associated with a Majoranafermionic operator γ_(j) at the jth position. The operators γ_(j) mayeach obey the fermionic anticommutation relation {γ_(j),γ_(k)}=2δ_(jk).For any ordered pair of MZMs 32 j and k, their joint fermionic parityoperator is given by iγ_(j)γ_(k)=−iγ_(k)γ_(j), which has eigenvaluesp_(jk)=±1 for even and odd parity respectively. The fermionic parity ofan MZM 32 is a topological charge (also known as a fusion channel)included in the quantum state 22 when the topological quantum computingdevice 20 includes MZMs 32. The corresponding projection operator ontothe subspace with parity s=p_(jk)=±1 is given by equation 200 in FIG. 5.In addition, the joint fermionic parity operator iγ_(j)γ_(k) isexpressed in terms of the positive and negative projection operators inequation 202. As further shown in FIG. 5, the joint fermionic parityoperator and the even-parity and odd-parity projection operators can beexpressed in a diagrammatic calculus as the diagram elements 204, 206,and 208 respectively. The joint fermionic parity operator diagramelement 204 may be expressed as a wavy line between two straight linesrepresenting the states of MZMs j and k. The joint fermionic parityoperator diagram element 204 may also be written as an antisymmetriccombination of the projectors 206 and 208.

In FIG. 6, equation 210 shows the basis states for a Majorana hexon 30.The ground states of the Majorana hexon 30 are assumed to have a fixedvalue of the collective fermionic parity due to the charging energy ofthe Majorana hexon 30. In the example of FIG. 6, the ground states ofthe Majorana hexon 30 each have an even collective fermionic parityp₁₂p₃₄p₅₆=+1. Thus, states with odd collective parity are excited statesassociated with quasiparticle poisoning. In this way, the low-energystate space of the hexon is 4-dimensional, with the basis states shownin equation 210. Thus, the Majorana hexon 30 may be viewed as atwo-qubit system with a first qubit encoded by p₃₄ and a second qubitencoded in p₁₂ in which the basis states shown in equation 210 are |0,0

, |0,1

, |1,0

, and |1,1

respectively. The joint fermionic parity operators may be expressed interms of Pauli operators on the two qubits as shown in equation 212,where the Pauli matrices are shown in equation 214.

According to the convention used herein, the third MZM 32C and thefourth MZM 32D of the Majorana hexon 30A are used as ancillary MZMs withjoint parity p₃₄=+1, and the computational qubit is encoded in p₁₂=p₅₆.The basis states of the computational qubit are |0

=|p₁₂=p₅₆=+

and |1

=|p₁₂=p₅₆=−

. The designation of the ancillary MZMs 32C and 32D and thecomputational MZMs 32A, 32B, 32E, and 32F is representeddiagrammatically in diagram 216 of FIG. 7. A general logical qubit statemay be represented by the diagram 218 in FIG. 7, in which the logicalqubit state is represented as a weighted sum of basis states.

A single-qubit Clifford gate may be implemented on the computationalqubit in a topologically protected manner via a “measurement-only”braiding protocol. The braiding transformations may be represented interm of Majorana operators as shown in equation 220 of FIG. 8. Equation220 shows a counterclockwise exchange of MZMs 32 at positions j and k.Using the measurement-only protocol, the single-qubit Clifford gate maybe realized by sequentially measuring the joint fermionic parities ofMZM pairs. These sequential measurements may be subject to the followingconstraints. First, the first measurement involves exactly one MZM 32from the ancillary pair. Second, subsequent measurements involve exactlyone MZM 32 from the preceding measured pair. Third, the finalmeasurement involves the original ancillary pair and the measurementoutcome equals the ancillary pair's initial joint parity, which is takento be p₃₄=+1 as discussed above. Thus, sequential measurementscorrespond to anti-commuting parity operators, where measurements ofpairs (jk) and (lm) are allowed to follow one another if and only ifiγ_(j)γ_(k)iγ_(l)γ_(m)=iγ_(l)γ_(m)iγ_(j)γ_(k). Each measurement sequencemay be viewed as a sequence of anyonic teleportations where, in eachmeasurement, the encoded qubit state is re-encoded in a different set ofMZMs 32, and where the measured pair of MZMS 32 temporarily becomes theancillary pair. In this view, the sequence of teleportations defines thebraiding “path” and enacts the corresponding braiding transformation onthe encoded state.

In some embodiments, the topological quantum computing device 20 may beconfigured to implement the logic gate 40 at least in part by performinga forced measurement of a projection operator 42 included in the logicgate 40. In such embodiments, the forced measurement may includeperforming a first measurement of a joint fermionic parity operatoriγ_(j)γ_(k) of the quantum state 22, as discussed above. Performing theforced measurement may further include determining whether a result ofthe first measurement is a predetermined target value 44. For example,the predetermined target value 44 may be the initial joint parity of theancillary pair. When the result of the first measurement is not thepredetermined target value 44, performing the forced measurement mayfurther include repeating a second measurement 52 that was performedprior to the first measurement 52 in the measurement sequence 50.Repeating the second measurement 52 may reset the quantum state 22 to astate prior to the first measurement 52 in the measurement sequence 50.Performing the forced measurement may further include repeating thefirst measurement 52 of the joint fermionic parity operator. Thus, thejoint fermionic parity operator iγ_(j)γ_(k) may be repeatedly measureduntil it is measured to be equal to the predetermined target value 44.

A sequence of projectors on a Majorana hexon 30 subject to the aboveconstraints may generate a single-qubit Clifford gate acting on theencoded computational qubit. FIG. 8 further shows an example of ameasurement sequence 50 in equation 222, where S is the π/4 phase gate.The initial measurement of the ancillary pair shown in equation 222 isredundant when the ancillary MZMs 32 are initialized with the positiveparity. As another example, equation 224 shows the gate B=S^(†)HS^(†),where H is the Hadamard gate, acting on the qubit. The gate set {S, B}is a generating set for all single-qubit Clifford gates C₁.

FIG. 9 shows an example diagram 226 of a measurement sequence 50 thatproduces the π/4 phase gate S discussed above. FIG. 9 further shows anexample diagram 228 of a measurement sequence 50 that produces the Bgate. The initial measurements of the ancillary pair are omitted in bothdiagram 226 and diagram 228, since the parity of the ancillary pair isassumed to be initialized to p₃₄=+1.

Two-qubit gates may be generated from sequences of 2-MZM and/or 4-MZMprojection operators. Turning now to FIG. 10, an ancillary projectionoperator for a two-qubit operation is defined in equation 230. Thetwo-qubit operation may be performed at two Majorana hexons 30A and 30B.Here, the MZMs 32G, 32H, 32I, 32J, 32K, and 32L included in the secondMajorana hexon 30B are labeled 1′ through 6′ to distinguish them fromthe MZMs 32A, 32B, 32C, 32D, 32E, and 32F included in the first Majoranahexon 30A. FIG. 10 further shows, in equation 232, a definition of theancillary projection operator for operations on more than two qubits.

In addition, FIG. 10 shows a definition for a parity operator inequation 234. In equation 234, M is a set of 2N MZMs 32 of N Majoranahexons 30, including two MZMs 32 from each Majorana hexon 30. Since theparticle number on each Majorana hexon 30 is preserved, each measurementoperator involves an even number of Majorana operators on each Majoranahexon 30. FIG. 10 further shows a definition of a projection operator onthe set M. In equation 236, s=±1.

The ancillary projection operator may begin each measurement sequence 50when a two-qubit operation is performed. In addition, the ancillaryprojection operator may end each measurement sequence 50 when thetwo-qubit operation is performed. In the case of two-qubit operations,each measurement sequence 50 may also end with the ancillary projectionoperator so that both ancillary pairs end up in their respectiveinitialized states. However, if Π₊ ⁽³⁴⁾ or Π₊ ^((3′4′)) commutes withevery term in the measurement sequence 50, then the final measurement ofthe respective ancillary pairs does not need to involve thecorresponding measurement pairs of MZMs 32, since the measurement pairswill already be in the final ancillary state.

A system of N Majorana hexons 30 may encode N computational qubits,which is a space of dimension 2^(N). The general condition for ameasurement sequence 50 of fermionic parity measurements involving NMajorana hexons 30 to compile to a unitary gate acting on thecomputational qubits is that the measurements 52 (which may range from2-MZM to 2N-MZM measurements) do not read information out of thecomputational state. Thus, the projection operators included in themeasurement sequence 50 do not collapse the encoded computational state.The measurement sequence 50 therefore may not include any projectorsequences that multiply out to an operator of rank less than 2^(N).

The relation between projector sequences and logic gates 40 ismany-to-one. A specific sequence of projectors, corresponding to aspecific sequence of measurements and outcomes (or forced-measurements),used to generate a gate with a measurement-only protocol may be denotedas shown in equation 238 of FIG. 11. In equation 238, the labels M_(μ)are used to denote an allowed set of an even number of MZMs 32 whosejoint fermionic parity is projected onto corresponding parity s_(μ) atthe μth projector in the sequence. The resulting unitary gate G actingon the encoded computational state space is shown in equation 240 ofFIG. 11.

In embodiments in which the logic gate 40 is constructed from twoMajorana hexons 30A and 30B, the Hilbert space of the two Majoranahexons 30A and 30B is the tensor product of that of the two Majoranahexons 30A and 30B. FIG. 12 shows, in equation 242, the quantum state 22formed by the tensor product of the respective states of the firstMajorana hexon 30A and the second Majorana hexon 30B.

In order to generate entangling two-qubit gates, one or moremeasurements 52 may be made of the collective fermionic parity of fourMZMs 32. The 4-MZM joint parity projector is shown in equation 244 ofFIG. 12, where the measurement 52 is performed on the MZMs 32 labeled jand k from the first Majorana hexon 30A and the MZMs 32 labeled l′ andm′ from the second Majorana hexon 30B. Similarly to the 2-MZM jointparity projector shown in equation 200, the 4-MZM joint parity projectorof equation 236 does not change the total fermionic parity of eitherMajorana hexon 30A or 30B.

Equation 246 of FIG. 12 shows an example of a two-qubit entangling gateW. A sequence of projectors that may be measured to obtain the gate W isshown in equation 248. Depending on the outcomes of measuring theprojection operators shown in equation 248, either W may be obtained(when s₁s₂s₃=+1) or the inverse of W (when s₁s₂s₃=−1) may be obtained.The first term in the tensor product acts on the ancillary qubits andthe second acts on the computational qubits. Since Π₊ ^((3′4′)) commuteswith every operator included in the measurement sequence 50 of equation248, the final projector only acts on the ancillary pair of MZMs 32.Equation 250 shows an expansion of the sequence of projection operatorsshown in equation 248. In equation 250, the ancillary projectionoperator Π₊ ^((3′4′)) is factored out, since it commutes with each ofthe other projection operators in the measurement sequence 50.

The gate set {S, B, W}, where the single-qubit gates can act on anyqubit and the two-qubit gates can act on any nearest-neighbor pair ofqubits, may generate any N-qubit Clifford gate C_(N). For example, thecontrolled-Z gate may be obtained as C(Z)=(S^(†)⊗S^(†))W, and thecontrolled not gate may be obtained from C(Z) by conjugating the secondqubit by H=SBS. Thus, since {S, B, C(Z)} generates the entire set ofN-qubit Clifford gates for any N, the gate set {S, B, W} also does.

As discussed above, some measurements 52 may be more difficult toperform than other measurements 52. These differences in measurementdifficulty may include differences in error rates of the measurements52. Additionally or alternatively, other factors may be used todetermine the difficulty of a measurement 52. The error rate of ameasurement 52 may be affected by a distance between the MZMs 32 onwhich the measurement 52 is performed. In some embodiments, thedifficulty of a measurement 52 may be lower when the MZMs 32 beingmeasured are closer to each other in the lattice. The processor 12 maybe configured to assign respective estimated weighted resource costs 54to the measurements 52 to account for these differences in difficulty.

In the two-sided Majorana hexon architecture 80 of FIG. 2 and theone-sided Majorana hexon architecture 82 of FIG. 3, measurements 52 areperformed by coupling MZMs 32 to quantum dots 78. FIGS. 14A-C show threeexample fermionic parity measurement configurations 300, 302, and 304.In the examples of FIGS. 14A-C, the couplings between the Majoranahexons 30 and the quantum dots 78 form interference loops delineated bythe paths connecting the MZMs 32 through the Majorana hexon 30 and thepaths connecting MZMs 32 through the quantum dots 78. To select theinterference paths, electrostatic depletion gates are provided such thatdifferent parts of the semiconductor may be connected or disconnected.These gates are referred to as cutter gates 76.

Each cutter gate 76 that is opened to form the semiconductor quantum dotconfiguration may increase the difficulty, as the number of open cuttergates 76 used to realize a quantum dot 78 affects the size of thequantum dot 78. When the number of open cutter gates 76 increases, thecoherence of the quantum dot 78 may decrease, thus adding a source ofnoise to the measurement. In addition, the overall length of thesemiconducting path may affect phase coherence, and the volume of thesemiconducting region enclosed by the path may affects properties of thequantum dot 78 such as its charging energy and level spacing.Measurements are typically easier to perform for smaller quantum dots78.

The plurality of cutter gates 76 included in a semiconducting path mayinclude one or more vertical cutter gates 76A. The plurality of cuttergates 76 may further include one or more horizontal cutter gates 76B. Inthe fermionic parity measurement configuration 302 of FIG. 14A, thesemiconducting path includes two vertical cutter gates 76A; in thefermionic parity measurement configuration 304 of FIG. 14B, thesemiconducting path includes seven vertical cutter gates 76A; and in thefermionic parity measurement configuration 306 of FIG. 14C, thesemiconducting path includes one vertical cutter gate 76A. The length ofthe semiconducting path may, in some embodiments, be proportional to thenumber of vertical cutter gates 76A included in the semiconducting path.

Wherever an MZM 32 couples to the semiconductor 74, the coupling may betuned by a cutter gate 76 forming a tunnel junction 330. In contrast tocutter gates 76 between semiconducting regions, which are typicallyeither fully opened or closed, each tunnel junction 330 may be tunedsuch that the ratio of the coupling energy of the tunnel junction 330 tothe charging energy E_(C) of the MZM 32 is in a regime in which theeffect of the MZM state on the quantum dot 78 is quickly and reliablymeasurable, while not suppressing the charging energy of the quantum dot78 or increasing the probability of quasiparticle poisoning. Typically,the visibility of the signal is reduced with each additional tunneljunction 330. Each tunnel junction 330 between an MZM 32 and a quantumdot 78 involved in a measurement 52 has an associated tunnelingamplitude. An decrease in the tunneling amplitude may decrease thevisibility of the measurement, since decreasing the tunneling amplitudemay make the energy splitting between states smaller. This decrease inthe visibility of the measurement 52 may increase the measurement timeand/or decrease the precision of the measurement 52.

In addition, noise in the tunneling junction 330 may interfere with themeasurement signal. As part of the measurement protocol, the couplingbetween the MZM 32 and the semiconductor 74 may be tuned from zero toits target value on a timescale that is shorter than the timescale ofthe measurements 52 but slower than a timescale at which diabaticcorrections would occur.

The number of tunnel junctions 330 involved in a measurement 52 may beequal to the number of MZMs 32 involved in the measurement 52, since atunnel junction 330 may be provided for each MZM 32. The fermionicparity measurement configuration 302 of FIG. 14A includes two tunneljunctions 330, the fermionic parity measurement configuration 304 ofFIG. 14B includes four tunnel junctions 330, and the fermionic paritymeasurement configuration 306 of FIG. 14C also includes four tunneljunctions 330.

Fluctuations of the background magnetic field may be another source ofnoise for the measurement. The contribution of these fluctuations to thenoise may be proportional to the area enclosed by the interferometricloop delineated by the architecture of the topological quantum computingdevice 20 and the geometry of a given measurement 52. In someembodiments, the hexon architecture geometries may be such that therelevant areas for such errors may be approximately partitioned intointeger multiples of a unit area 340. In the fermionic paritymeasurement configuration 302 of FIG. 14A, the semiconducting path hasan area of two times the unit area 340; in the fermionic paritymeasurement configuration 304 of FIG. 14B, the semiconducting path hasan area of seven times the unit area 340; and in the fermionic paritymeasurement configuration 306 of FIG. 14C, the semiconducting path hasan area of three times the unit area 340.

The difficulty of a measurement 52 may also depend on the number ofMajorana hexons 30 involved in the measurement 52. This is because themeasurement visibility may be affected by how precisely the quantumstate 22 may be tuned to the degenerate tunneling point. In addition,the operations utilized in a measurement 52 may cause errors thattransfer MZMs 32 between the different Majorana hexons 30. Increasingthe number of Majorana hexons 30 involved in a measurement 52 mayincrease the probability of such errors.

Given the factors described above, an estimated weighted resource cost54 of a 2N-MZM measurement 52 involving N Majorana hexons 30 is shown inFIG. 14A in equation 306. In equation 306, n_(c) is the number ofvertical cutter gates 76 opened for the measurement, n_(a) is an areaenclosed by the interferometry loop delineated by the measurement 52(expressed as an integer multiple of a unit area), and n_(t) is thenumber of tunneling junctions involved in the measurement 52. Thequantities w_(c), w_(a), and w_(t) are weighting factors respectivelyassociated with n_(c), n_(a), and n_(t). The weights w_(t) associatedwith the tunneling junctions may also include contributions from thehorizontal cutter gates 76B, since the horizontal cutter gates 76B maybe used to control tunneling. The effect of the number N of Majoranahexons 30 on the estimated weighted resource cost 54 is denoted as f(N).Each of w_(c), w_(a), w_(t), and f(N) may be determined experimentallyfor the particular topological quantum computing device 20 at which themeasurements 52 are performed.

FIGS. 15A-E show example fermionic parity measurement configurations310, 312, 314, 316, and 318 that may be used with the one-sided Majoranahexon architecture 82 shown in FIG. 3. The fermionic parity measurementconfiguration 310 of FIG. 15A has three vertical cutter gates 76A, twotunneling junctions 330, and an interference loop enclosing an area ofthree times the unit area 340. The fermionic parity measurementconfiguration 312 of FIG. 15B has three vertical cutter gates 76A, fourtunneling junctions 330, and an interference loop enclosing an area offive times the unit area 340. The fermionic parity measurementconfiguration 314 of FIG. 15C has zero vertical cutter gates 76A, fourtunneling junctions 330, and an interference loop enclosing an area oftwo times the unit area 340. The fermionic parity measurementconfiguration 316 of FIG. 15D has two vertical cutter gates 76A, eighttunneling junctions 330, and an interference loop enclosing an area ofsix times the unit area 340. The fermionic parity measurementconfiguration 318 of FIG. 15E has four vertical cutter gates 76A, eighttunneling junctions 330, and an interference loop enclosing an area offour times the unit area 340. The example fermionic parity measurementconfigurations 312, 314, 316, and 318 of FIGS. 15B-E respectively areconfigurations of corresponding measurements 52 in an upward direction,a downward direction, a rightward direction, and a leftward direction.

So far, the MZMs 32 in a Majorana hexon 30 have been labeled 1, . . . ,6 and have been assigned roles in the measurement 52 according to theselabels. For example, in the computational basis, the MZMs 32 labeled as3 and 4 serve as the ancillary pair, while MZMs 1, 2, 5, and 6collectively encode the computational qubit. However, the six labels maybe assigned to the physical MZMs 32 of a Majorana hexon 30 according toother labeling schemes. The choice of labeling scheme may affect thedifficulty of a measurement 52, as discussed below. In some embodiments,the processor 12 may be configured to determine the estimated totalresource cost 56 of at least one measurement sequence 50 at least inpart by relabeling the MZMs 32 included in the Majorana hexon 30 tochange which MZMs 32 are included in the computational qubit and whichMZMs 32 are included in the ancillary qubit. In embodiments in which thetopological quantum computing device 20 includes one or more Majoranatetrons 130 or Majorana octons, the MZMs 32 included in the Majoranatetrons 130 or Majorana octons may also be relabeled change which MZMs32 are included in the computational qubit and which MZMs 32 areincluded in the ancillary qubit.

In the following example, let (a, b, c, d, e, f) denote theconfiguration of MZMs 32 within a Majorana hexon 30, where for one-sidedMajorana hexons, the labeling goes from top to bottom as shown in FIG.3, and for two-sided Majorana hexons, the labeling goes counterclockwisefrom the top-left to the top-right as shown in FIG. 2. A possibleconfiguration for either Majorana hexon architecture, as discussedabove, is

1, 2, 3, 4, 5, 6

. Here, MZMs 1 and 6 are on opposite ends of the Majorana hexon 30. Onthe other hand, in the configuration

1, 6, 2, 3, 4, 5

, the MZMs 1 and 6 are adjacent. Thus, different configurations of MZMs30 will result in different assignments of estimated weighted resourcecosts 54 to measurements 52. For example, a measurement of MZMs 1 and 6may have estimated weighted resource costs 54 of

and

, the latter of which may be higher. If this measurement 52 occursfrequently in a measurement sequence 52, it may be lessresource-intensive to use the configuration

1, 6, 2, 3, 4, 5

. In addition, measurements 52 involving pairs that are neighbors indifferent directions may have different estimated weighted resourcecosts 54. For example, in a one-sided Majorana hexon architecture 82,measurements 52 connecting vertical neighbors may be lessresource-intensive than measurements 52 connecting horizontal neighbors.

The two-sided Majorana hexon architecture 80 and the one-sided Majoranahexon architecture 82 each have symmetry relations that may reduce thenumber of labeling configurations to evaluate. A two-sided Majoranahexon has horizontal and vertical reflection symmetry, which reduces thenumber of inequivalent configurations from 6!=720 to 180. One-sidedMajorana hexons have vertical reflection symmetry, which reduces thenumber of inequivalent configurations from 720 to 360.

In order for implementation of the logic gate 40 to be scalable, thefull array of Majorana hexons 30 in the topological quantum computingdevice 20 may utilize labeling configurations that are periodic in thearray. In some embodiments, each Majorana hexon 30 in the array may usethe same labeling configuration. However, different configurations maybe assigned to different Majorana hexons 30 in other embodiments. Forexample, the array may include a first configuration for allright-facing one-sided Majorana hexons 30 and a second configuration forall left-facing one-sided Majorana hexons 30.

In some embodiments, the topological quantum computing device 20 may beconfigured to implement the logic gate 40 at least in part by performinga forced measurement of a projection operator 42 included in the logicgate 40. The forced measurement may include performing a firstmeasurement 52 of a joint fermionic parity operator Γ_(M) of the quantumstate 22. When the joint fermionic parity operator Γ_(M) of an orderedset of M MZMs 32 is measured in a system in a pure state |Ψ

, the measurement outcome s=± may be obtained with the probability shownin equation 400 in FIG. 16A. A post-measurement state shown in equation402 is obtained when this measurement 52 is made. For general statesdescribed by a density matrix ρ, the measurement outcome s may beobtained with the probability shown in equation 404. Thepost-measurement state for general states described by the densitymatrix ρ is shown in equation 406.

The forced measurement may further include determining whether a resultof the first measurement 52 is a predetermined target value 44. Sincethe predetermined target value 44 of s is not always obtained when thefirst measurement 52 is performed, the projection operator 42 with thetarget parity is not always obtained. In order to obtain the targetprojection operator Π_(s) ^((jk)) in the measurement-only scheme, arepeat-until-success “forced measurement” procedure may be used. When ameasurement of iγ_(j)γ_(k), the joint fermionic parity operator of thetarget projection operator Π_(s) ^((jk)), is performed, the probabilityof obtaining the target parity may be ½. The initial measurement 52 ofthe ancillary pair of MZMs 32 may have a deterministic outcome. When theresult of the first measurement 52 of iγ_(j)γ_(k) is not thepredetermined target value 44, the forced measurement may furtherinclude resetting the quantum state 22 to a state prior to the firstmeasurement 52. The quantum state 22 may be reset by performing a paritymeasurement on the pair of MZMs measured in a second measurement 52 thatwas performed prior to the first measurement 52.

Subsequently to resetting the quantum state 22, performing the forcedmeasurement may further include repeating the first measurement 52 ofthe joint fermionic parity operator iγ_(j)γ_(k). If the repetition ofthe first measurement 52 also does not return the predetermined targetvalue 44 of the parity, the steps above may be repeated. Thus, the jointfermionic parity operator iγ_(j)γ_(k) may be measured and reset untilthe measurement 52 returns the predetermined target value 44.

FIG. 16B shows a flowchart of an example method 408 by which the forcedmeasurement may be performed. At step 410, the method 408 may includeperforming a measurement 52 of a joint fermionic parity operator of thequantum state. At step 412, the method 408 may further includedetermining whether a result of the measurement 52 is a predeterminedtarget value 44 of the parity. If the result of the measurement is notthe predetermined target value 44, the method 408 may include, at step414, resetting the quantum state 22. The method 408 may then return tostep 410. If the result of the measurement 52 is the predeterminedtarget value 44, the method 408 may further include, at step 416, usinga projection operator with the predetermined target value 44 in themeasurement sequence 50.

An example in which forced measurement is used to obtain a measurementsequence 50 is shown in FIGS. 17A-C. In the example of FIGS. 17A-C, theS gate is generated with the projection operator sequence Π₊ ⁽³⁴⁾Π₊⁽²³⁾Π₊ ⁽¹³⁾. A diagram 418 for this projector sequence is shown in FIG.17A. In the example of FIGS. 17A-C, the measurement of iγ₁γ₃ results inthe target value 44 of s₁=+, but the measurement of iγ₂γ₃ results in theundesired value of s₂=−, as shown in diagram 420 in FIG. 17B. As shownin diagram 422 of FIG. 17C, the measurement of iγ₁γ₃ may be repeated.The quantum state 22 is reset regardless of the outcome of this repeatedmeasurement. The measurement of iγ₂γ₃ may then be repeated, with another½ probability of obtaining the predetermined target result s₂=−. If theundesired measurement outcome is obtained again, the above steps may berepeated until the predetermined target outcome is obtained. Themeasurement sequence 50 depicted in diagram 422 differs only by anoverall phase from the measurement sequence 50 depicted in diagram 424,as indicated in equation 426.

In order to distinguish the application of a forced-measurementoperation from the projectors associated with a physical measurement,the application of this forced-measurement to the MZM pair (jk) in asequence following a measurement of (kl) is denoted in equation 428 ofFIG. 18. In equation 428, the desired measurement outcome s is obtainedat the nth attempt. In addition, s_(a)≠s for a=1, . . . , n−1, and themeasurement outcomes r_(a) are irrelevant.

The estimated total resource cost 56 of the forced measurement sequenceof equation 428 may be given by equation 430 in some embodiments. Theestimated total resource cost 56 provided in equation 430 is equal tothe estimated total resource cost 56 of an average-case measurementsequence 50 including

n

=2 attempts.

As an alternative to the forced measurement procedure discussed above,the following procedure may be used to obtain the predetermined targetvalue 44 of a measurement 52. When the measurement 52 of the MZM pair(jk) following a measurement 52 of the MZM pair (kl) yields an undesiredoutcome, instead of resetting the quantum state 22 by repeating theprevious measurement 52 of (kl), the quantum state 22 may instead bereset by measuring the MZM pair (jl). Thus, resetting the quantum state22 may include, in the alternative forced measurement protocol,measuring an MZM pair including a first MZM 30 (labeled above as j) onwhich the first measurement 52 was performed. The MZM pair furtherincludes a second MZM 30 (labeled above as l) on which a secondmeasurement 52 that was performed prior to the first measurement 52 inthe measurement sequence 50 was performed. However, unlike in the firstforced measurement protocol discussed above, the second measurement 52was not performed on the first MZM 30 in the MZM pair.

In more general terms not specific to topological quantum computingdevices 20 that include MZMs 32, resetting the quantum state 22according to the alternative forced measurement protocol includesmeasuring a plurality of topological charges. The plurality oftopological charges including a symmetric difference of a firstplurality of topological charges on which the first measurement 52 wasperformed and a second plurality of topological charges on which asecond measurement 52 was performed prior to the first measurement 52 inthe measurement sequence 50. “Symmetric difference” is defined as theunion of two sets minus the intersection of the two sets.

The alternative forced measurement approach is shown diagrammatically inFIGS. 19A-B for the projector sequence Π₊ ⁽³⁴⁾Π₊ ⁽³⁶⁾Π₊ ⁽²³⁾Π₊ ⁽¹³⁾,when an undesired measurement outcome occurs for the measurement of MZMs(36). FIG. 19A shows, at diagram 432, the projector sequence after theundesired measurement outcome s₃=− has occurred. FIG. 19A further showsa first forced measurement sequence in diagram 434. FIG. 19B shows, indiagram 436, a second forced measurement sequence that may be used as analternative to that of FIG. 19A. In addition, as shown in equation 438in FIG. 19B, the projector sequence shown in diagram 436 differs only byan overall constant from the projector sequence Π₊ ⁽³⁴⁾Π₊ ⁽³⁶⁾Π₊ ⁽²³⁾Π₊⁽¹³⁾.

In order to differentiate the alternative forced-measurement protocolfrom the first forced measurement protocol (and from an ordinaryprojector), this alternative forced measurement protocol as applied tothe MZM pair (jk) following a measurement 52 of the MZM pair (kl) isdefined in equation 440 of FIG. 20. In equation 440, s_(a)≠s for a=1, .. . , n−1, and the measurement outcomes p_(a) are irrelevant. As in thefirst forced measurement protocol discussed above, the estimated totalresource cost 56 of the forced measurement sequence of equation 440 maybe the geometric mean of the weighted estimated resource costs 54 of themeasurements 52 in the measurement sequence 50. This estimated totalresource cost 56 may be equal to the difficulty weight of theaverage-case measurement sequence 50 including

n

=2 attempts. The estimated total resource cost 56 in such embodiments isgiven in equation 442 of FIG. 20. This alternative forced measurementprotocol may be less resource intensive than the first forcedmeasurement protocol in embodiments where parity measurements of MZMs(jl) have lower weighted estimated resource costs 54 than measurementsof MZMs (kl).

Forced measurement protocols for 2N-MZM measurements, in particular4-MZM measurements, are discussed below with reference to FIGS. 21-22.These forced measurement protocols may also be used for 2-MZMmeasurements that follow a 4-MZM measurement. Equation 444 of FIG. 21shows the condition under which a forced measurement of M₂ follows ameasurement of M₁ for some choice of M₃. As another condition forperforming the forced measurement on M₂, the subsequent projectionoperators in the measurement sequence 50 may not commute with eachother. This condition holds when Γ_(M) ₁ Γ_(M) ₂ =−Γ_(M) ₂ Γ_(M) ₁ andΓ_(M) ₂ Γ_(M) ₃ =−Γ_(M) ₃ Γ_(M) ₂ .

Another form of the left-hand side of equation 444 is shown in equation446 of FIG. 21. Equation 448 further shows the conditions under whichequation 444 holds. Under the conditions given in equation 448, Γ_(M) ₃may be replaced with a constant. FIG. 21 provides a generalization ofthe two different forced measurement protocols described above. Thesecond condition shown in equation 448 may lead to invalid measurementsequences that would collapse the qubit states or to measurements ofgreater than 2N-MZMs if M₁ and M₂ include more than two elements each.Since the estimated total resource cost 56 grows quickly as the numberof MZMs 32 increases, it is preferable to avoid having to measure suchsequences. However, the above problems do not arise when the firstcondition of equation 448 is satisfied.

Equation 450 of FIG. 22 shows four example forced measurement protocolsfor measurement sequences 50 involving four MZMs 32. In addition,equation 452 shows the respective estimated total resource costs 56 ofthe measurement sequences 50 of equation 450, according to oneembodiment.

The forced measurement protocols discussed above provides control overwhich fermionic parities are projected upon at each measurement 52 inthe measurement sequence 50. These forced measurement protocols allowfor the implementation of a projector sequence that generates aspecified target logic gate 40. A forced measurement protocol could beapplied for every projector in a given projector sequence. However, sucha strategy may be inefficient, since the different projectors in themeasurement sequence 50 may have a correlated effect on the resultinglogic gate 40. When determining the estimated total resource cost 56 fora measurement sequence, the processor 12 may be further configured todetermine which projectors in a measurement sequence 50 have acorrelated effect and, therefore, which specific measurements 52 cantolerate any outcome and which measurements 52 may have to be forced inorder to obtain the logic gate 40. In such embodiments, the processor 12may be configured to determine the respective estimated total resourcecost 56 of each measurement sequence 50 at least in part by identifyingone or more measurement sequences 50 that differ by an overall Paulioperator, as discussed in further detail below.

Returning to FIG. 6, equation 212 shows the joint fermionic parityoperators iγ_(j)γ_(k) expressed in terms of the Pauli operators shown inequation 214. In addition, as shown in FIG. 5, a projection operatorΠ_(s) ^((jk)) may be drawn as a cap and cup, connected by a wavy line ifs=− and unconnected if s=+. For every s=− projector in the measurementsequence, the corresponding fermion line (that terminates on two MZMlines) may be moved to the top of the diagram using the diagrammaticrules. Each such fermion line that has been slid to the top of thediagram simply connects two MZM lines j and k, resulting in a jointfermionic parity operator iγ_(j)γ_(k). In embodiments in which everymeasurement sequence starts and ends with a forced measurement of π₊^((anc)), the fermion lines do not connect to the ancillary MZM lineswhen pushed to the top of the diagram. Thus, in such embodiments, j andk do not correspond to ancillary MZMs 32.

Turning now to FIG. 23, the fermion lines that are slid to the top ofthe diagram correspond to the Pauli operators shown in table 500. Thecomplete operation effected on the computational subspace by ameasurement sequence 50 may be a braiding transformation (hence aClifford gate) determined by which MZMs 32 are measured in themeasurement sequence 50, followed by a Pauli gate determined by themeasurement outcomes. Equation 502 of FIG. 23 shows a single-hexonprojector sequence that compiles to a Clifford gate G. In equation 502,the projection channel parities s_(μ) need not all be positive. Equation504 shows the single-hexon projector sequence of equation 502 whenrespective parities of all the projection operators have been set topositive. Equation 502 may be rewritten as equation 506, where q is thenumber of projection operators with negative parities in equation 502, pis an integer, and P is a Pauli gate. Thus, the effect of themeasurement outcomes s_(μ) in a single-hexon projector sequence is tochange the resulting logic gate 40 by at most a Pauli gate.

FIG. 24A shows, in equation 508, a projection operator sequence that maybe used to implement any of the Pauli gates. In equation 508, theresulting Pauli gate P is independent of s₁ and s₄. This independence isshown in FIGS. 24A-B with the diagram 510, which differs only by overallphases from the diagrams 512 and 514 of FIG. 24B. Isotopy of the MZMlines allows them to be straightened, leaving no nontrivial braiding indiagram 514. Thus, P₊=1, where P₊ is the Pauli gate that results whenall measurement outcomes s_(μ)=+ in equation 508. In addition, in FIGS.24A-B, both ends of the s₁ line connect to the j=5 MZM line whenstraightened and both ends of the s₄ line connect to the j=1 MZM linewhen straightened, which allows the s₁ and s₄ lines to be removedwithout affecting the resulting logic gate 40. After straightening theMZM lines and sliding the s_(μ) lines to the top of the diagram, s₂=−contributes the operator iγ₂γ₅=Z⊗X, s₃=− contributes the operatoriγ₁γ₂=1⊗X, and s₅=− contributes the operator iγ₁γ₂=1⊗Z. Thus, thecompiled Pauli gate is P as shown in equation 508.

For multi-hexon projector sequences, changing the projection channelparities s_(μ) also changes the resulting logic gate 40 by at most amulti-qubit Pauli gate. In addition, by tracking the effects of theprojection channel parities s_(μ) on the resulting compiled gate, theprocessor 12 may be configured to determine one or more measurements 52in the measurement sequence 50 for which to perform forced measurements.For a single-hexon projector sequence, when all the fermion lines in theprojector sequence are moved to the top of the diagram, each line mayeither be removed or may end up in one of the six configurationsconnecting MZM lines represented by the fermion parity operatorsiγ_(j)γ_(k) listed in table 500. In this way, the specific Paulioperator contributed by a given measurement outcome to P in thedecomposition shown in equation 506 may be determined. Therefore, anyClifford gate may be generated from a measurement sequence 50 in whichthree or fewer of the measurements 52 in the measurement sequence 50 areforced measurements. Of these three or fewer forced measurements, oneforced measurement may be used to assign a positive parity to theancillary pair of MZMs 32, and the other two or fewer forcedmeasurements may be used to obtain a target Pauli gate. For example, themeasurement sequence 50 of FIGS. 24A-B may generate a particular targetPauli gate for any values of s₁, s₃, and s₄ by choosing s₂ and s₅appropriately via forced measurements.

Another approach that may be used in addition to, or as an alternativeto, forced measurement in measurement-only topological quantum computingis described below. In this approach, known as Majorana-Pauli tracking,measurement outcomes that only change the resulting braidingtransformations by Pauli gates may be tracked. More generally, a similartracking strategy may be employed when the measurement outcomes areAbelian anyons. Majorana-Pauli tracking may allow for the use of fewerphysical measurement operations and may allow deterministic measurementsequences 50 to be used for topological gate operations.

FIG. 25 shows the construction of a measurement sequence 50 thatcompiles to a gate G acting on the computational state space in a systemof N Majorana hexons 32. First, in equation 516, an ancillary projectionoperator 516 is defined for the N-hexon system. In addition, equation518 introduces a reversal operator that flips the state of eachancillary qubit whose initial and final projections differ. In equation518, γ_(a,j) is the ath MZM 32 of the jth Majorana hexon 30. In equation520, the ancillary projection operator of equation 516 is expressed interms of the reversal operator of equation 518. Equation 522 providesthe measurement sequence 522 by which the gate G is compiled accordingto the Majorana-Pauli tracking protocol, in terms of the operatorsdefined in equations 516, 518, and 520. The operator G shown in equation522 is a unitary operator and therefore does not reduce the rank of thecomputational subspace.

When Majorana-Pauli tracking is used, instead of the convention p₃₄=+1used with the forced measurement protocol, p₃₄ may take either apositive or negative value. The value of p₃₄ may change over the courseof generating a measurement-only gate. As in the forced measurementprotocol, the computational qubit is encoded in p₁₂. Thus, when thecollective fermionic parity of the Majorana hexon 30 is even, theremaining parity p₅₆ is determined by the respective parities of theother two pairs according to p₅₆=p₁₂p₃₄. For the general case in whichthe Majorana hexon 30 may have a collective fermionic parity p_(h) thatmay be even or odd, the parity p₅₆ is given by p₅₆=p₁₂p₃₄p_(h).

As shown in equation 524, an N-qubit Pauli operator may be applied tothe gate G of equation 522 to produce the same gate with a differentsequence of projections. Thus, if a measurement-only sequence ofmeasurements 52 is performed to obtain a logic gate 40 and themeasurement outcomes are tracked, the resulting logic gate 40 will havea known Pauli gate correction. If the non-Clifford gates that areutilized in a quantum computation are single-qubit phase gates (in anyof the Pauli bases), the Pauli gate correction may be pushed through thephase gates with at most a Clifford gate correction. When a Cliffordgate correction is performed, a Clifford gate by which the logic gate 40differs from the target logic gate may be tracked as the measurementsequence 50 is performed. In such embodiments, the Clifford gatecorrection may be dealt with by updating the subsequent Clifford gate inthe computation. When non-Clifford phase gates are implemented byinjecting states, such a Clifford correction would typically be used.Thus, the effect of the Clifford correction on performance would besmall.

Approaches are discussed below by which the processor 12 may be furtherconfigured to determine a first measurement sequence 60 that has alowest estimated total resource cost 66 of the plurality of measurementsequences 50. The lowest estimated total resource cost 66 may be aglobal minimum estimated total resource cost 66 across all measurementsequences 50 that implement the logic gate 40 or may alternatively be aminimum estimated total resource cost of a subset of all suchmeasurement sequences 50 that are searched by the processor 12.

When a topological quantum computation is performed using a plurality ofdifferent logic gates 40, the processor 12 may be configured to selectthe lowest estimated total resource cost 66 for a subset of theplurality of different logic gates 40. Within a given topologicalquantum computation, it is not always possible for all logic gates 40involved in the computation to be implemented with their respectivefirst measurement sequences 60. In some embodiments, the firstmeasurement sequence 60 may be used for one or more logic gates 40 thatare used with a high frequency in the topological quantum computation.For example, the one or more logic gates 40 for which the firstmeasurement sequence 60 is determined may be the controlled-Pauli gates,the Hadamard gate, or all single-qubit Clifford gates.

When the processor 12 searches for the first measurement sequence 60,the processor 12 may perform the search over measurement sequences 50used in the Majorana-Pauli tracking protocol or the forced measurementprotocol. Determination of the first measurement sequence 60 having thelowest estimated total resource cost 66 is first discussed herein forthe Majorana-Pauli tracking protocol. When a measurement sequence 50 iscompiled in terms of the projector sequence shown in equation 522, thesequence of physical measurements that will be performed is the sequenceM₁, . . . , M_(n) specified in the projector sequence. When the physicalmeasurement outcomes do not match the specified projector channelss_(μ), the resulting gate differs from G by at most a Pauli gate, whichmay be tracked and compensated for at a later time. As such, thismeasurement-only realization of G may be assigned the estimated totalresource cost 56 shown in equation 526 of FIG. 26.

When Majorana-Pauli tracking is utilized, Clifford gates may be groupedinto Pauli cosets given by collections of Clifford gates that areequivalent up to multiplication by an overall multi-qubit Pauli gate.The Pauli coset of an N-qubit Clifford gate G is defined in equation 528of FIG. 26. When using Majorana-Pauli tracking, it is not necessary togenerate every Clifford gate. Instead, one Clifford gate from each Paulicoset may be generated, since differences by Pauli gates are dealt withby the tracking protocol. Thus, in some embodiments, the most easilyrealized Clifford gate in a given Pauli coset may be used to implementthe entire class of Clifford gates. Thus, the element of each Paulicoset with the lowest estimated total resource cost 66 may be used whenany Clifford gate in that Pauli coset is called in a computation.

When a forced measurement protocol is utilized instead, the measurementsequence 50 for compiling a gate G may be written as shown in equation530 of FIG. 27. When determining the first measurement sequence 60 withthe lowest estimated total resource cost 66, the processor 12 may beconfigured to determine the minimum number of projectors for whichforced measurement is needed to generate the gate G with the forcedmeasurement protocol. The projector sequence may then be converted intoa measurement sequence 50 by utilizing forced measurement for eachprojector where forced measurement is needed. Standard measurements maybe performed for the other projectors. For each projector for whichforced measurement is performed, the forced measurement protocol that isused, of the two forced measurement protocols discussed above, may bethe forced measurement protocol for which the estimated weightedresource cost 54 of the forced measurement is lower.

The estimated total resource cost 56 of a measurement sequence 50 when aforced measurement protocol is used may be determined by taking thegeometric mean of the possible total resource costs of the measurementsequence 50. As shown in equation 532 of FIG. 27, this geometric meanmay be obtained by starting with the expression from equation 430 forthe difficulty weight of the projector sequence and replacing theweights of the projectors that are forced with the average difficultyweight corresponding to the forced-measurement protocol used. Inequation 532, F₁ is the set of projectors in the sequence to beimplemented by forced measurements of the first type and F₂ is the setof projectors in the sequence to be implemented by forced measurementsof the second type.

When searching for measurement sequences 50 that may be used toimplement a logic gate 40, the processor 12 may be configured todetermine what measurement sequences 50 do not collapse thecomputational state. For single-qubit gates, such measurement sequences50 may satisfy the condition that consecutive 2-MZM measurements musthave exactly one MZM 32 in common. Under such a condition, eachmeasurement step may involve choosing one MZM 32 from the previousmeasurement pair and one from the four remaining MZMs 32, leading toeight possible measurements 52 to choose from. The nth measurement 52 inthe measurement sequence 50 may be constrained to be a measurement ofthe ancillary pair (3, 4) of MZMs 32. In addition, the second-to-lastmeasurement 52 may be constrained to involve one MZM 32 of the precedingpair and one MZM 32 of the ancillary pair. Thus, there may be fourchoices available for the second-to-last measurement 52. The size of thesearch space for single-hexon measurement sequences of length n may be2^(3n−4). Even though this scaling is exponential in n, the value of nin the first measurement sequence 60 with the lowest estimated totalresource cost 66 is typically low for single-qubit gates.

Once the one or more measurement sequences 50 that produce a targetlogic gate 40 without collapsing the computational state are determined,the resulting logic gates G may be evaluated for all possiblemeasurement outcomes s_(μ). In some embodiments, the processor 12 may beconfigured to perform a brute-force search by determining a respectiveestimated total resource cost 54 for each measurement sequence 50 thatimplements a target logic gate 40 and is shorter than a predeterminedlength 58. For example, the predetermined length 58 may be n=9.

The example provided below discusses determination of the firstmeasurement sequence 60 and the lowest estimated total resource cost 66for the set of controlled Pauli gates {C(X), C(Y), C(Z)} and the SWAPgate, which are examples of two-qubit Clifford gates. A quantum state 22of two Majorana hexons 30 has 510 different nontrivial fermionic parityprojectors. In some embodiments, each of the nontrivial fermionic parityprojectors may be tested at each measurement step to determine whichprojectors do not collapse the computational state. In such embodiments,the processor 12 may be configured to diagonalize the projector sequenceafter each time a projector is added. When the projector sequence hasbeen diagonalized, the processor 12 may be further configured to discardthe projector sequence if the projector sequence collapses thecomputational state. However, performing the diagonalization of theprojector sequence may be computationally costly.

As an alternative, for small values of n, each of the 510 possibleparity projectors may be applied at each step in the projector sequence.The final resulting logic gates 40 generated by the projector sequencesmay then be checked. In the example in which the respective firstmeasurement sequences 60 are determined for each of the controlled Pauligates, projector sequences that produce the controlled Pauli gates mayinclude at least four projectors. In this example, a search of eachmeasurement sequence 50 up to the predetermined length 58 of n=4 may beperformed.

Additionally or alternatively, each possible projector sequence, up tosome predetermined length 58, that includes one 4-MZM projector may besearched for the controlled Pauli gates. For example, the predeterminedlength 58 may be n=5.

Each projector sequence that compiles to the SWAP gate includes at leasttwo 4-MZM projectors. A search for the two 4-MZM projectors may beperformed for projector sequences up to the predetermined length 58 ofn=4. For the one-sided Majorana hexon architecture 82, no projectorsequences that compile to the SWAP gate are found for this predeterminedlength 58. Instead, a projector sequence that compiles to the SWAP gatemay be formed from a plurality of controlled not gates.

For a two-qubit measurement sequence 50 that compiles to a target logicgate 40, the measurement sequence 50 may be evaluated for each possibleprojector channel s_(μ). Evaluating each measurement sequence 50 foreach possible projector channel s_(μ) may provide the Pauli correctiongate that may be used in the Majorana-Pauli tracking protocol and mayalso identify the projectors for which forced measurement is performedin the forced measurement protocol.

Correlations between the remaining measurement outcomes may then beidentified. Such correlations may be determined starting with the firstprojector that does not have fixed projection channel, denoted as s_(v).The subsets of projector sequences where s_(v)=+1 and where s_(v)=−1 maybe considered separately. Within each subset, the processor 12 may beconfigured to check whether any subsequent measurement 50 has a fixedoutcome. If a subsequent measurement 50 has a fixed outcome, thatmeasurement 50 may be forced onto a channel that is correlated withs_(v). If the measurement 50 does not have a fixed outcome, the abovesteps may be recursively applied with the projection channel of thatmeasurement replacing s_(v).

In one example of compiling a two-qubit gate from a measurement sequence50, the sequence Π₊ ⁽³⁴⁾Π_(s) ₃ ⁽³⁵⁾Π_(s) ₂ ⁽⁵⁶⁾Π_(s) ₁ ^((35; 1′6′))complies to a controlled not gate when s₂=+ and s₃=s₁. Thus, theprojector sequences Π₊ ⁽³⁴⁾Π₊ ⁽³⁵⁾Π₊ ⁽⁵⁶⁾Π₊ ^((35; 1′6′)) and Π₊ ⁽³⁴⁾Π⁻⁽³⁵⁾Π₊ ⁽⁵⁶⁾Π⁻ ^((35; 1′6′)) yield the same logic gate 40. In thisexample, forced measurement may be performed for μ=2, 3, 4.

In one example of identifying the first measurement sequence 60, theexample weighting factors w_(c)=1.25, w_(a)=1.01, w_(t)=1.65, andf(N)=(N!)^(N−1) may be used in equation 306. In this example, wheneither forced measurement or Majorana-Pauli tracking is used with atwo-sided Majorana hexon architecture 80, the MZM labeling configuration

3, 4, 1, 2, 6, 5

yields the lowest estimated total resource cost 66 for each of thesingle-qubit Hadamard gate, the geometric mean of all single-qubitClifford gates, the geometric mean of controlled not gates acting in allfour directions, and the geometric mean of controlled Pauli gates actingin all four directions. For one-sided Majorana hexon architectures 82,when forced measurement is used, the MZM labeling configuration

1, 2, 6, 3, 4, 5

yields the lowest estimated total resource cost 66 for the Hadamard gateand the geometric mean of all single-qubit Clifford gates, and the MZMlabeling configuration

3, 4, 1, 2, 6, 5

yields the lowest estimated total resource cost 66 for the geometricmean of controlled not gates acting in all four directions and thegeometric mean of controlled Pauli gates acting in all four directions.For one-sided Majorana hexon architectures 82, when Majorana-Paulitracking is used, the MZM labeling configuration

1, 2, 6, 3, 4, 5

yields the lowest estimated total resource cost 66 for each of thesingle-qubit Hadamard gate, the geometric mean of all single-qubitClifford gates, the geometric mean of controlled not gates acting in allfour directions, and the geometric mean of controlled Pauli gates actingin all four directions.

FIG. 28 shows a flowchart of an example method 600 for performing aquantum computation by implementing a logic gate. The method 600 may beused with the quantum computing system 10 of FIG. 1, the example quantumcomputing system 100 of FIG. 4, or some other quantum computing system.At step 602, the method 600 may include identifying a plurality ofmeasurement sequences that implement a logic gate. Each measurementsequence may include a plurality of measurements of a quantum state of atopological quantum computing device. In some embodiments, thetopological quantum computing device may include at least one of aMajorana tetron including four MZMs, a Majorana hexon including sixMZMs, or a Majorana octon including eight MZMs. The topological quantumcomputing device may include a plurality of Majorana tetrons, hexons,and/or octons in some embodiments. Additionally or alternatively, thetopological quantum computing device may include a Majorana tetronincluding four MZMs and/or a Majorana octon including eight MZMs. Ameasurement sequence may include measurements of a plurality of Majoranatetrons, hexons, and/or octons in some embodiments.

In some embodiments, step 602 may include, at step 604, identifying oneor more measurement sequences that implement the logic gate multipliedby an overall Pauli operator. When two measurement sequences differ byan overall Pauli operator, the Pauli operator may be tracked as themeasurements in the measurement sequence are performed when thetopological quantum computing device implements the logic gate. Acorrection for the overall Pauli operator may be performed at the end ofthe measurement sequence. Step 604 may allow computational resources tobe saved by reducing the number of measurement sequences that arechecked by more computationally intensive methods.

At step 606, the method 600 may further include determining a respectiveestimated total resource cost of each measurement sequence of theplurality of measurement sequences. For each measurement sequence of theplurality of measurement sequences, step 606 may include, at step 608,determining an estimated weighted resource cost of each measurementincluded in the measurement sequence. In some embodiments, the estimatedweighted resource cost of each measurement may indicate an error rate ofthe measurement. In embodiments in which step 608 is performed, step 606may further include, at step 610, determining the estimated totalresource cost of the measurement sequence based on the plurality ofestimated weighted resource costs. For example, the estimated totalresource cost may be a product of the estimated weighted resource costsof the measurements. In some embodiments, step 606 may includedetermining a respective estimated total resource cost for eachmeasurement sequence that implements the logic gate and is shorter thana predetermined length.

In embodiments in which the quantum state includes a Majorana tetron,Majorana hexon, or Majorana octon, the MZMs included in the Majoranatetron, Majorana hexon, or Majorana octon may have a labeling order thatindicates a plurality of MZMs included in a computational qubit and aplurality of MZMs included in an ancillary qubit. In such embodiments,step 606 may further include, at step 612, modifying a topologicalencoding of a computational qubit and an ancillary qubit. Thesetopological encodings may be modified by relabeling the MZMs included inthe Majorana tetron, Majorana hexon, or Majorana octon to change how thecomputational and ancillary qubits are encoded in the physical MZMs.

At step 614, the method 600 may further include determining a firstmeasurement sequence that has a lowest estimated total resource cost ofthe plurality of measurement sequences. At step 616, the method 600 mayfurther include implementing the logic gate at the topological quantumcomputing device by applying the first measurement sequence to thequantum state. In some embodiments, step 616 may include the steps ofthe example method 408 for performing a forced measurement shown in FIG.16B. In embodiments in which step 604 is performed, step 616 may furtherinclude, at step 618, tracking a Pauli gate correction to the logic gateas the measurement sequence is performed.

Although the above examples are provided for topological quantumcomputing devices 20 including Majorana hexons 30, the systems andmethods discussed above may be used when performing topological quantumcomputation with other non-Abelian anyons or defects. In suchembodiments, measurements of fusion channels other than fermionicparities may be performed. The forced measurement and Majorana-Paulitracking protocols may be used when the measurement outcomes correspondto fusion channels that are Abelian. In addition to MZMs, examples ofstructures that may be used in the topological quantum computing deviceinclude Ising anyons and Parafendleyons (parafermionic zero modes).Although the topological charge discussed in the examples provided aboveis a joint fermionic parity, the topological charge may be some otherquantity when defects other than MZMs are included in the topologicalquantum computing device. Forced measurement protocols may also be usedwhen the measurement outcomes correspond to non-Abelian fusion channels.In addition, the resource cost estimation systems and methods describedabove may be applied to other measurement-based operations, such as theinjection of non-Clifford gates.

In some embodiments, the methods and processes described herein may betied to a computing system of one or more computing devices. Inparticular, such methods and processes may be implemented as acomputer-application program or service, an application-programminginterface (API), a library, and/or other computer-program product.

FIG. 29 schematically shows a non-limiting embodiment of a computingsystem 700 that can enact one or more of the methods and processesdescribed above. Computing system 700 is shown in simplified form.Computing system 700 may embody the quantum computing system 10described above and illustrated in FIG. 1. Computing system 700 may takethe form of one or more personal computers, server computers, tabletcomputers, home-entertainment computers, network computing devices,gaming devices, mobile computing devices, mobile communication devices(e.g., smart phone), and/or other computing devices, and wearablecomputing devices such as smart wristwatches and head mounted augmentedreality devices.

Computing system 700 includes a logic processor 702 volatile memory 704,and a non-volatile storage device 706. Computing system 700 mayoptionally include a display subsystem 708, input subsystem 710,communication subsystem 712, and/or other components not shown in FIG.29.

Logic processor 702 includes one or more physical devices configured toexecute instructions. For example, the logic processor may be configuredto execute instructions that are part of one or more applications,programs, routines, libraries, objects, components, data structures, orother logical constructs. Such instructions may be implemented toperform a task, implement a data type, transform the state of one ormore components, achieve a technical effect, or otherwise arrive at adesired result.

The logic processor may include one or more physical processors(hardware) configured to execute software instructions. Additionally oralternatively, the logic processor may include one or more hardwarelogic circuits or firmware devices configured to executehardware-implemented logic or firmware instructions. Processors of thelogic processor 702 may be single-core or multi-core, and theinstructions executed thereon may be configured for sequential,parallel, and/or distributed processing. Individual components of thelogic processor optionally may be distributed among two or more separatedevices, which may be remotely located and/or configured for coordinatedprocessing. Aspects of the logic processor may be virtualized andexecuted by remotely accessible, networked computing devices configuredin a cloud-computing configuration. In such a case, these virtualizedaspects are run on different physical logic processors of variousdifferent machines, it will be understood.

Non-volatile storage device 706 includes one or more physical devicesconfigured to hold instructions executable by the logic processors toimplement the methods and processes described herein. When such methodsand processes are implemented, the state of non-volatile storage device706 may be transformed—e.g., to hold different data.

Non-volatile storage device 706 may include physical devices that areremovable and/or built-in. Non-volatile storage device 706 may includeoptical memory (e.g., CD, DVD, HD-DVD, Blu-Ray Disc, etc.),semiconductor memory (e.g., ROM, EPROM, EEPROM, FLASH memory, etc.),and/or magnetic memory (e.g., hard-disk drive, floppy-disk drive, tapedrive, MRAM, etc.), or other mass storage device technology.Non-volatile storage device 706 may include nonvolatile, dynamic,static, read/write, read-only, sequential-access, location-addressable,file-addressable, and/or content-addressable devices. It will beappreciated that non-volatile storage device 706 is configured to holdinstructions even when power is cut to the non-volatile storage device706.

Volatile memory 704 may include physical devices that include randomaccess memory. Volatile memory 704 is typically utilized by logicprocessor 702 to temporarily store information during processing ofsoftware instructions. It will be appreciated that volatile memory 704typically does not continue to store instructions when power is cut tothe volatile memory 704.

Aspects of logic processor 702, volatile memory 704, and non-volatilestorage device 706 may be integrated together into one or morehardware-logic components. Such hardware-logic components may includefield-programmable gate arrays (FPGAs), program- andapplication-specific integrated circuits (PASIC/ASICs), program- andapplication-specific standard products (PSSP/ASSPs), system-on-a-chip(SOC), and complex programmable logic devices (CPLDs), for example.

The terms “module,” “program,” and “engine” may be used to describe anaspect of computing system 700 typically implemented in software by aprocessor to perform a particular function using portions of volatilememory, which function involves transformative processing that speciallyconfigures the processor to perform the function. Thus, a module,program, or engine may be instantiated via logic processor 702 executinginstructions held by non-volatile storage device 706, using portions ofvolatile memory 704. It will be understood that different modules,programs, and/or engines may be instantiated from the same application,service, code block, object, library, routine, API, function, etc.Likewise, the same module, program, and/or engine may be instantiated bydifferent applications, services, code blocks, objects, routines, APIs,functions, etc. The terms “module,” “program,” and “engine” mayencompass individual or groups of executable files, data files,libraries, drivers, scripts, database records, etc.

When included, display subsystem 708 may be used to present a visualrepresentation of data held by non-volatile storage device 706. Thevisual representation may take the form of a graphical user interface(GUI). As the herein described methods and processes change the dataheld by the non-volatile storage device, and thus transform the state ofthe non-volatile storage device, the state of display subsystem 708 maylikewise be transformed to visually represent changes in the underlyingdata. Display subsystem 708 may include one or more display devicesutilizing virtually any type of technology. Such display devices may becombined with logic processor 702, volatile memory 704, and/ornon-volatile storage device 706 in a shared enclosure, or such displaydevices may be peripheral display devices.

When included, input subsystem 710 may comprise or interface with one ormore user-input devices such as a keyboard, mouse, touch screen, or gamecontroller. In some embodiments, the input subsystem may comprise orinterface with selected natural user input (NUI) componentry. Suchcomponentry may be integrated or peripheral, and the transduction and/orprocessing of input actions may be handled on- or off-board. Example NUIcomponentry may include a microphone for speech and/or voicerecognition; an infrared, color, stereoscopic, and/or depth camera formachine vision and/or gesture recognition; a head tracker, eye tracker,accelerometer, and/or gyroscope for motion detection and/or intentrecognition; as well as electric-field sensing componentry for assessingbrain activity; and/or any other suitable sensor.

When included, communication subsystem 712 may be configured tocommunicatively couple various computing devices described herein witheach other, and with other devices. Communication subsystem 712 mayinclude wired and/or wireless communication devices compatible with oneor more different communication protocols. As non-limiting examples, thecommunication subsystem may be configured for communication via awireless telephone network, or a wired or wireless local- or wide-areanetwork, such as a HDMI over Wi-Fi connection. In some embodiments, thecommunication subsystem may allow computing system 700 to send and/orreceive messages to and/or from other devices via a network such as theInternet.

According to one aspect of the present disclosure, a computing system isprovided, including a processor configured to identify a plurality ofmeasurement sequences that implement a logic gate. Each measurementsequence may include a plurality of measurements of a quantum state of atopological quantum computing device. The processor may be furtherconfigured to determine a respective estimated total resource cost ofeach measurement sequence of the plurality of measurement sequences. Theprocessor may be further configured to determine a first measurementsequence that has a lowest estimated total resource cost of theplurality of measurement sequences. The topological quantum computingdevice may be configured to implement the logic gate by applying thefirst measurement sequence to the quantum state.

According to this aspect, for each measurement sequence of the pluralityof measurement sequences, the processor may be configured to determinethe respective estimated total resource cost at least in part bydetermining an estimated weighted resource cost of each measurementincluded in the measurement sequence. The processor may be furtherconfigured to determine the estimated total resource cost of themeasurement sequence based on the plurality of estimated weightedresource costs.

According to this aspect, the estimated weighted resource cost of eachmeasurement may indicate an error rate of the measurement.

According to this aspect, the processor may be configured to determine arespective estimated total resource cost for each measurement sequencethat implements the logic gate and is shorter than a predeterminedlength.

According to this aspect, the processor may be configured to determinethe first measurement sequence at least in part by modifying atopological encoding of a computational qubit and an ancillary qubit.

According to this aspect, the topological quantum computing device mayinclude a plurality of Majorana zero modes (MZMs).

According to this aspect, the topological quantum computing device mayinclude at least one of a Majorana tetron including four MZMs, aMajorana hexon including six MZMs, or a Majorana octon including eightMZMs.

According to this aspect, the topological quantum computing device mayinstantiate the plurality of MZMs in a one-sided architecture or atwo-sided architecture.

According to this aspect, the processor may be configured to identifythe plurality of measurement sequences that implement the logic gate atleast in part by identifying one or more measurement sequences thatimplement the logic gate multiplied by an overall Pauli operator.

According to this aspect, the processor may be further configured totrack a Pauli gate correction to the logic gate when the topologicalquantum computing device implements the logic gate.

According to this aspect, the topological quantum computing device maybe configured to implement the logic gate at least in part by performinga forced measurement of a projection operator. The forced measurementmay include performing a first measurement of a topological charge ofthe quantum state. The forced measurement may further includedetermining whether a result of the first measurement is a predeterminedtarget value. When the result of the first measurement is not thepredetermined target value, the forced measurement may further includeresetting the quantum state and repeating the first measurement of thetopological charge.

According to this aspect, resetting the quantum state may includerepeating a second measurement that was performed prior to the firstmeasurement in the measurement sequence.

According to this aspect, resetting the quantum state includes measuringa plurality of topological charges including a symmetric difference of afirst plurality of topological charges on which the first measurementwas performed and a second plurality of topological charges on which asecond measurement was performed prior to the first measurement in themeasurement sequence.

According to another aspect of the present disclosure, a method forperforming a quantum computation is provided. The method may includeidentifying a plurality of measurement sequences that implement a logicgate. Each measurement sequence may include a plurality of measurementsof a quantum state of a topological quantum computing device. The methodmay further include determining a respective estimated total resourcecost of each measurement sequence of the plurality of measurementsequences. The method may further include determining a firstmeasurement sequence that has a lowest estimated total resource cost ofthe plurality of measurement sequences. The method may further includeimplementing the logic gate at the topological quantum computing deviceby applying the first measurement sequence to the quantum state.

According to this aspect, for each measurement sequence of the pluralityof measurement sequences, determining the respective estimated totalresource cost may include determining an estimated weighted resourcecost of each measurement included in the measurement sequence. Theestimated total resource cost of the measurement sequence may bedetermined based on the plurality of estimated weighted resource costs.The estimated weighted resource cost of each measurement may indicate anerror rate of the measurement.

According to this aspect, the topological quantum computing device mayinclude a plurality of Majorana zero modes (MZMs).

According to this aspect, the topological quantum computing device mayinstantiate the plurality of MZMs in a one-sided architecture or atwo-sided architecture. The topological quantum computing device mayinclude at least one of a Majorana tetron including MZMs, a Majoranahexon including six MZMs, or a Majorana octon including eight MZMs.

According to this aspect, identifying the plurality of measurementsequences that implement the logic gate may include identifying one ormore measurement sequences that implement the logic gate multiplied byan overall Pauli operator.

According to this aspect, implementing the logic gate may includeperforming a forced measurement of a projection operator included in thelogic gate. The forced measurement may include performing a firstmeasurement of a topological charge of the quantum state. The forcedmeasurement may further include determining whether a result of thefirst measurement is a predetermined target value. When the result ofthe first measurement is not the predetermined target value, the forcedmeasurement may further include resetting the quantum state andrepeating the first measurement of the topological charge.

According to another aspect of the present disclosure, a computingsystem is provided, including a processor configured to identify aplurality of measurement sequences that implement a logic gate. Eachmeasurement sequence may include a plurality of measurements of aquantum state of a topological quantum computing device. The topologicalquantum computing device may include a Majorana hexon including sixMajorana zero modes (MZMs). The processor may be further configured todetermine an estimated weighted resource cost of each measurementincluded in each measurement sequence of the plurality of measurementsequences. For each measurement sequence, the processor may be furtherconfigured to determine the estimated total resource cost of themeasurement sequence based on the estimated weighted resource costs ofthe measurements included in the measurement sequence. The processor maybe further configured to determine a first measurement sequence that hasa lowest estimated total resource cost of the plurality of measurementsequences. The topological quantum computing device may be configured toimplement the logic gate by applying the first measurement sequence tothe quantum state.

It will be understood that the configurations and/or approachesdescribed herein are exemplary in nature, and that these specificembodiments or examples are not to be considered in a limiting sense,because numerous variations are possible. The specific routines ormethods described herein may represent one or more of any number ofprocessing strategies. As such, various acts illustrated and/ordescribed may be performed in the sequence illustrated and/or described,in other sequences, in parallel, or omitted. Likewise, the order of theabove-described processes may be changed.

The subject matter of the present disclosure includes all novel andnon-obvious combinations and sub-combinations of the various processes,systems and configurations, and other features, functions, acts, and/orproperties disclosed herein, as well as any and all equivalents thereof.

The invention claimed is:
 1. A computing system comprising: a processorconfigured to: identify a plurality of measurement sequences thatimplement a logic gate, each measurement sequence including a pluralityof measurements of a quantum state of a topological quantum computingdevice; for each measurement sequence of the plurality of measurementsequences that implements the logic gate and is shorter than apredetermined number of measurements, determine a respective estimatedtotal error rate of implementing the logic gate at the quantum stateusing that measurement sequence; and determine a first measurementsequence that has a lowest estimated total error rate of the pluralityof measurement sequences, wherein the topological quantum computingdevice is configured to implement the logic gate by applying the firstmeasurement sequence to the quantum state.
 2. The computing system ofclaim 1, wherein, for each measurement sequence of the plurality ofmeasurement sequences, the processor is configured to determine therespective estimated total error rate at least in part by: determiningan estimated weighted error rate of each measurement included in themeasurement sequence; and determining the estimated total error rate ofthe measurement sequence based on the plurality of estimated weightederror rates.
 3. The computing system of claim 1, wherein the processoris configured to determine the first measurement sequence at least inpart by modifying a topological encoding of a computational qubit and anancillary qubit.
 4. The computing system of claim 1, wherein thetopological quantum computing device includes a plurality of Majoranazero modes (MZMs).
 5. The computing system of claim 4, wherein thetopological quantum computing device includes at least one of a Majoranatetron including four MZMs, a Majorana hexon including six MZMs, or aMajorana octon including eight MZMs.
 6. The computing system of claim 4,wherein the topological quantum computing device instantiates theplurality of MZMs in a one-sided architecture or a two-sidedarchitecture.
 7. The computing system of claim 4, wherein the processoris configured to identify the plurality of measurement sequences thatimplement the logic gate at least in part by identifying one or moremeasurement sequences that implement the logic gate multiplied by anoverall Pauli operator.
 8. The computing system of claim 7, wherein theprocessor is further configured to track a Pauli gate correction to thelogic gate when the topological quantum computing device implements thelogic gate.
 9. The computing system of claim 1, wherein the topologicalquantum computing device is configured to implement the logic gate atleast in part by performing a forced measurement of a projectionoperator, wherein the forced measurement includes: performing a firstmeasurement of a topological charge of the quantum state; determiningwhether a result of the first measurement is a predetermined targetvalue; when the result of the first measurement is not the predeterminedtarget value: resetting the quantum state; and repeating the firstmeasurement of the topological charge.
 10. The computing system of claim9, wherein resetting the quantum state includes repeating a secondmeasurement that was performed prior to the first measurement in themeasurement sequence.
 11. The computing system of claim 9, whereinresetting the quantum state includes measuring a plurality oftopological charges including a symmetric difference of: a firstplurality of topological charges on which the first measurement wasperformed; and a second plurality of topological charges on which asecond measurement was performed prior to the first measurement in themeasurement sequence.
 12. A method for performing a quantum computation,the method comprising: identifying a plurality of measurement sequencesthat implement a logic gate, each measurement sequence including aplurality of measurements of a quantum state of a topological quantumcomputing device; for each measurement sequence of the plurality ofmeasurement sequences that implements the logic gate and is shorter thana predetermined number of measurements, determining a respectiveestimated total error rate of implementing the logic gate at the quantumstate using that measurement sequence; determining a first measurementsequence that has a lowest estimated total error rate of the pluralityof measurement sequences; and implementing the logic gate at thetopological quantum computing device by applying the first measurementsequence to the quantum state.
 13. The method of claim 12, wherein, foreach measurement sequence of the plurality of measurement sequences,determining the respective estimated total error rate includes:determining an estimated weighted error rate of each measurementincluded in the measurement sequence; and determining the estimatedtotal error rate of the measurement sequence based on the plurality ofestimated weighted error rates.
 14. The method of claim 12, wherein thetopological quantum computing device includes a plurality of Majoranazero modes (MZMs).
 15. The method of claim 14, wherein: the topologicalquantum computing device instantiates the plurality of MZMs in aone-sided architecture or a two-sided architecture; and the topologicalquantum computing device includes at least one of a Majorana tetronincluding MZMs, a Majorana hexon including six MZMs, or a Majorana octonincluding eight MZMs.
 16. The method of claim 14, wherein identifyingthe plurality of measurement sequences that implement the logic gateincludes identifying one or more measurement sequences that implementthe logic gate multiplied by an overall Pauli operator.
 17. The methodof claim 12, wherein implementing the logic gate includes performing aforced measurement of a projection operator included in the logic gate,wherein the forced measurement includes: performing a first measurementof a topological charge of the quantum state; determining whether aresult of the first measurement is a predetermined target value; whenthe result of the first measurement is not the predetermined targetvalue: resetting the quantum state; and repeating the first measurementof the topological charge.
 18. A computing system comprising: aprocessor configured to: identify a plurality of measurement sequencesthat implement a logic gate, each measurement sequence including aplurality of measurements of a quantum state of a topological quantumcomputing device, wherein the topological quantum computing deviceincludes a Majorana hexon including six Majorana zero modes (MZMs);determine an estimated weighted error rate of each measurement includedin each measurement sequence of the plurality of measurement sequencesthat implements the logic gate and is shorter than a predeterminednumber of measurements; and for each measurement sequence of theplurality of measurement sequences, determine a respective estimatedtotal error rate of the measurement sequence based on the estimatedweighted error rates of the measurements included in the measurementsequence; and determine a first measurement sequence that has a lowestestimated total error rate of the plurality of measurement sequences,wherein the topological quantum computing device is configured toimplement the logic gate by applying the first measurement sequence tothe quantum state.